To present the ideal ROM, we first need to define the spaces of resolved ROM scales (i.e., \({X}^{r}\) ) and unresolved ROM scales (i.e., \({X}^{r^{\prime }}\) ).
To this end, we extend the variational multiscale (VMS) framework proposed by Hughes and his group two decades ago in the FEM context.
We note, however, that there are other ways of defining the spaces of resolved and unresolved ROM scales, e.g., spatial filtering [1]}.
